Introduction
Differentially describing groups of graphs lies at the heart of many scientific and societal challenges. Neuroscientists, for example, might want to characterize brain activity in healthy subjects, elucidate how it differs from brain activity in subjects diagnosed with certain disorders or diseases (e.g., autism or Alzheimer’s), and investigate whether their findings are the same across different groups of subjects (e.g., children, adolescents, and adults; or men and women). Policymakers, security experts, and epidemiologists alike could seek to understand patterns of human mobility, be it to improve the resilience of traffic infrastructure to random failures and targeted attacks, or to curb the spread of infectious diseases. And international economists might want to investigate patterns of world trade, e.g., imports and exports between countries, and ask how these vary across different years and product classes.
We refer to the common task underlying these scenarios as graph group analysis: Given a set of graphs and a partition of this set into graph groups, succinctly summarize the commonalities and differences between graphs in the same group, between graphs in different groups, and between the relationships connecting the groups. In this paper, we formalize graph group analysis as a maximum likelihood modeling problem, using significant subgraphs
as graph patterns to factorize our probability distribution. We introduce
Gragra (Graph group analysis) as an algorithm to solve this problem, which jointly discovers a set of graph patterns and an assignment of these patterns to graph groups.As a realworld example of graph group analysis, consider Fig. 1. Here, we show the top shared (left) and specific (right) patterns identified in restingstate functional brain networks of adolescents with and without autism spectrum disorder, where nodes in the graphs correspond to brain regions, and edges signal strong connectivity between regions. On the right, patterns with red edges are characteristic of autistic adolescents, and patterns with blue edges are characteristic of nonautistic adolescents. They indicate over and underconnectivity, respectively, in the brains of autistic adolescents when compared to typically developed controls. Although there is no consensus regarding the relationships between autism and neural connectivity Hull et al. (2017), our method identifies graph patterns that permit neuroscientific interpretation: For example, the dark blue pattern in Fig. 1 indicates underconnectivity between the visual cortex, responsible for processing visual information, and the lingual gyrus, involved in vision and word processing.
Graph group analysis is related to graph classification (e.g., Lee, Rossi, and Kong, 2018), but we are interested not only in what is different but also in what is similar among our graph groups. Our task further shares some of its motivation with significant subgraph mining (e.g., Sugiyama et al., 2015), graph summarization (e.g., Liu et al., 2018), and data clustering with graphs as data points (e.g., Mukherjee, Sarkar, and Lin, 2017). However, we focus on a complete characterization of a set of graphs under a given partition—a cornerstone of scientific discovery involving graph data.
The remainder of the paper is structured as follows. After settling our basic notation, we describe the theoretical foundations of our method and introduce our algorithm. Having covered related work, through experiments on synthetic and realworld data, we demonstrate that Gragra works well in practice, before rounding up with discussion and conclusions. All our data, code, and results are publicly available.^{1}^{1}1https://doi.org/10.5281/zenodo.5758560
Preliminaries
We consider a set of nodealigned graphs with nodes and edges, omitting the subscripts when clear from context. A partition is a set of nonempty subsets of , called graph groups, of cardinalities , whose disjoint union is . Our graphs can be undirected or directed, loopy or nonloopy, and unweighted, edgelabeled, or integer weighted, where for the purposes of our model, we treat distinct edge labels or edge weights as a set of categories, and edges are drawn from the set of all possible weighted edges.
The empirical frequency of edge set in group is , and we denote by the set of nodes incident with at least one edge in .
We base our probabilistic model on the maximum entropy principle, by which the distribution that best reflects a given set of constraints without introducing additional assumptions is the distribution with maximum Shannon entropy Jaynes (1982). Thus, the expected frequency of in under a given set of edge sets is
where is the power set of , and is the distribution maximizing , subject to linear constraints for all elements in Csiszár (1975). That is,
where and all are realvalued model parameters. Finding the distribution is a convex problem that involves computing the expected frequency over exponentially many elements. This is intractable if done naïvely, but there exist practical approaches that factorize into a product of independent distributions Mampaey, Vreeken, and Tatti (2012); Dalleiger and Vreeken (2020a, b).
Theory
We now lay the theoretical foundations of our method, introducing our probabilistic model, our objective function, and our statistical test. At a high level, our goal in graph group analysis is to discover a set of graph patterns, i.e., edge sets of connected subgraphs, and an association matrix assigning graph patterns to graph groups, such that and together reveal the similarities and differences between graphs in the same group, between graphs in different groups, and between the relationships connecting the groups. A pattern is specific if we assign it to only one graph group, and it is shared if we assign it to several graph groups. We choose which patterns to include in our model based on the information we gain from them, testing whether this gain is statistically significant to rule out spurious results.
To avoid redundancy, we assign a pattern to a group iff is informative for , given what we already know about all groups. More precisely, using as a column index of in a slight abuse of notation, we set iff is informative for under our current model . We assess this by comparing the empirical frequency of in group , , to its expected frequency in that group under our current model, , where , and is obtained from a practical approximation of the maximum entropy distribution with constraint set . is informative for iff is significantly different from , as judged by a statistical test, and we add to (and column to ) if is informative for some .
To identify a suitable set of graph patterns and an adequate association matrix , we exploit the interplay between two steps. First, we discover the best pattern to add to , given the current , and second, we identify the best assignment of to graph groups to update , given the current and a new pattern . We now describe each step in more detail.
Identifying Informative Graph Patterns
To measure the likelihood of a set of graph patterns, we use the Bayesian Information Criterion (BIC),
where is the number of coefficients in our model, and
is the log likelihood of (with derived from ), assuming that the graphs in a group are independent and identically distributed. This allows us to identify a good set of graph patterns by minimizing the BIC score, i.e.,
Solving this problem exactly poses significant challenges in practice due to its combinatorial nature and the explosion in the number of solution candidates. Therefore, we employ a greedy search strategy, iteratively selecting the graph pattern that best improves our current model. That is, for a given , we select the graph pattern that maximizes our likelihood, or equivalently, maximizes the difference , which we write as
In a nutshell, the core of our approach is the procedure
(1) 
by which we iteratively and greedily insert into the pattern that locally maximizes our information gain.
Using a model selection criterion alone, however, we cannot tell if our information gain is due to random fluctuations or due to signal, especially if we only have a limited number of samples. Thus, to avoid modeling noise, we add to only if its information gain is statistically significant
. Therefore, we test whether we can reject the null hypothesis
To this end, we use Vuong’s closeness test Vuong (1989)
, a likelihood ratio test designed for model selection problems under BIC. Vuong’s test statistic is defined as
, which is asymptotically distributed with degrees of freedom. To calculate , we count the coefficients that must be changed in every distribution if we insert into . As we add one coefficient for , and update at least edge coefficients per group, we arrive at additional degrees of freedom.Discovering Differential Pattern Associations
Once we have selected a new pattern to add to , given the current and , we identify a good assignment of to graph groups to update . Here, the significance of , which is used to accept into , signals that is informative for some , but it does not tell us for which . To assign to a graph group , we hence rely on the partial information gain of for ,
Again, we use Vuong’s closeness test to decide whether is significant; and if it is, we set .
Algorithm
Having stated its theoretical foundations, we now introduce Gragra as an algorithm to differentially describe groups of graphs using sets of significant subgraphs. Gragra, whose pseudocode is given as Alg. LABEL:alg:gragra, revolves around the procedure stated in Eq. (1), a greedy process that iteratively selects the graph pattern candidate that best enhances our model. This could involve myriad searches through the exponentiallysized space of all possible graphs with nodes from , which is not only computationally infeasible in most cases but also unnecessary, as most candidates will be eliminated as uninformative by Vuong’s test. Hence, rather than exhaustively searching for the best graph patterns, we propose to grow graphs by systematically adding edges to candidates.
algocf[t]
To enable our model to infer all possible graphs, we initialize it with the set of all possible edges. As our initial graph to grow, we then select the most promising graph pattern from our initial candidates, i.e., the connected triples
Starting with a graph pattern , we explore all its expansions, , from which we select the best candidate pattern to grow further, as long as we gain information and is significant. We summarize these steps in the function Grow of Alg. LABEL:alg:gragra (l. LABEL:l:grow:beginning–LABEL:l:grow:end).
Grow requires many inferences of , which involve inferring many more expected frequencies
, rendering exact computation impractical. We thus design a practical, pessimistic heuristic that only considers the information gain from graphs
in which is fully present:(2) 
We use instead of because it involves inferring only one expected frequency per graph group. A derivation of this heuristic can be found in the Online Appendix.
To summarize, Gragra proceeds as follows. Starting with an initial set of candidates (l. LABEL:l:candidates), we select (l. LABEL:l:select) and grow (l. LABEL:l:grow) the best candidate, and retain all significant expansions (l. LABEL:l:significant), until we have grown to its fullest potential (l. LABEL:l:fullpotentialcond–LABEL:l:fullpotential). Afterwards, we test if the information gain provided by is significant, and if so, we keep track of its graph group associations (l. LABEL:l:groupassoc), and insert into (l. LABEL:l:insert).
The computational complexity of Gragra depends on the number of candidates, which can grow to at most . In practice, Gragra’s complexity depends on the number of times we grow graph patterns, which is datadependent and bounded by the size of the largest connected component observed in an input graph, as growing beyond that reduces the information gain. Multiplying by the initial set of candidates, Gragra achieves a complexity of for all practial purposes, where we assume that the complexity of inferring the expected frequency is bounded.
Related Work
To the best of our knowledge, we are the first to differentially describe groups of graphs through sets of significant subgraphs. Our method is inspired by advances in graph similarity description (Momo, Coupette and Vreeken, 2021) and explainable pattern set mining using maximumentropy modeling (DISC, Dalleiger and Vreeken, 2020a, b). However, Momo focuses on pairs and unpartitioned sets of graphs; DISC is designed for itemset data, ignores graph structure, and does not scale on graphs; and neither method uses a statistical test to select patterns. Further related work broadly falls into two categories: statistical inference on network populations, and graph mining for groups of graphs.
Statistical Inference on Network Populations.
In the statistics literature, the task of analyzing multiple graphs simultaneously is typically framed as an inference problem for networkvalued random variables
Durante, Dunson, and Vogelstein (2017); Lovato et al. (2020); Lunagómez, Olhede, and Wolfe (2020). Here, Ghoshdastidar et al. (2020) establish limits for distinguishing two population distributions given small sample sizes, and Lunagómez, Olhede, and Wolfe (2020) propose notions of mean and dispersion for a single population of networks, where the population mean is itself a network. Maugis et al. (2020) use subgraph counts to test if all graphs in a sample are drawn from the same distribution, and Signorelli and Wit (2020) propose a modelbased clustering approach to describe subpopulations within a population of networks. Finally, Durante, Dunson, and Vogelstein (2017) extend latent space approaches designed for single graphs to capture the probabilistic mechanism that generates multiple graphs from a single population distribution. Their model has been used to characterize and test for differences between groups of brain networks Durante, Dunson et al. (2018)—an actively studied application for which numerous statistical methods, mostly focusing on testing for differences, have been developed Ginestet et al. (2017); Wang et al. (2019); Lukemire et al. (2020); Kundu et al. (2021); Lovato et al. (2021); Lehmann et al. (2021).Prior work in the statistics literature has focused on describing one network population or distinguishing two populations. In contrast, with Gragra, we aim to construct a differential description of any number of populations. Furthermore, we ask not only if these populations are different, but also how they are different and how they are similar.
Graph Mining for Groups of Graphs. In the graph mining literature, groups of graphs have been studied in contexts as diverse as significant subgraph mining LlinaresLópez et al. (2015); Sugiyama et al. (2015), graph classification Vogelstein et al. (2013); Yan et al. (2019); Lanciano, Bonchi, and Gionis (2020), graph clustering with graphs as data points Mukherjee, Sarkar, and Lin (2017), anomalous graph detection Gomes, Rao, and Neville (2018), and graph summarization for time series of graphs Shah et al. (2015). Significant subgraph mining commonly considers small, nodelabeled graphs with unaligned node sets, and hence, does not target our problem. However, our setup—i.e., mediumsized graphs with aligned node sets—, has received heightened attention in the graph classification community, again inspired by challenges from neuroscience Vogelstein et al. (2013); Yan et al. (2019); Lanciano, Bonchi, and Gionis (2020).
The methods that are closest to our work are contrast subgraphs Lanciano, Bonchi, and Gionis (2020) and signal subgraphs Vogelstein et al. (2013), both designed for two groups of nodealigned graphs. Contrast subgraphs discover the densest subgraph in the difference of the summary graphs of the input groups (obtained by adding the graphs in each group separately and then subtracting the results), where the size of this subgraph depends on a userspecified regularization parameter . Signal subgraphs assume edge independence as a prior to rank edges by the values of an edgewise statistical test for distributional difference (e.g., Fisher’s exact test). Like signal subgraphs, Gragra combines ideas from structural and statistical pattern mining to produce interpretable results that—unlike contrast subgraphs—are based on a statistical foundation. Gragra is more exploratory and more flexible than both competitors, however, because it treats graph group description as an end in itself and can handle any number of graph groups.
Experiments
We now present an extensive evaluation of our algorithm. To this end, we implement Gragra in C++ and expose a Python interface to facilitate experimentation. We run our experiments on Intel E52643 CPUs with or GB RAM, testing at a conservative significance level of (or when operating with less than samples), and make all data, code, and results publicly available.^{2}^{2}2https://doi.org/10.5281/zenodo.5758560 Our experiments revolve around two questions:

Can Gragra reliably recover the ground truth from groups of synthetic graphs?

Does Gragra discover meaningful patterns in groups of real graphs?
Recovering Ground Truth from Synthetic Graphs
To assess the reliability of Gragra, we run it on groups of synthetic graphs with planted patterns. We consider three scenarios, namely,

summarizing one group of graphs,

differentially describing two groups of graphs, and

differentially describing four groups of graphs.
In all three scenarios, each graph group consists of graphs with nodes, and our configurations differ in their planted patterns (type, prevalence, and position) and noise levels. A detailed overview of our synthetic data configurations can be found in the Online Appendix.
For each scenario, we report the distribution of precision, recall, and F1 score, computed separately for each group of graphs, for the edges of the planted patterns across graph group datasets sampled with different seeds. In all scenarios, we compare Gragra, which uses BIC with Vuong’s closeness test for pattern selection, with a variant using only BIC and no statistical test to select patterns (). For configurations in the second scenario, we also compare our results with those from contrast subgraphs (CSG) and signal subgraphs (SSG), described in the previous section.
As shown in Fig. 2,
delivers good results in the fourgroup scenario but generally has poor precision, treating noise as signal. CSG and SSG identify only constrastive patterns, and fail even for contrastive patterns if the individual edges in planted patterns have similar occurrence probabilities across groups.
Gragra, however, reliably recovers the ground truth across scenarios and configurations, which allows us to hope that it will also work in practice.Discovering Meaningful Patterns in Real Graphs
To determine whether Gragra discovers meaningful patterns in groups of real graphs, we run experiments on graph group data of various graph types from three domains: functional brain networks (undirected, unweighted), air transportation networks (directed, weighted), and international trade networks (directed, weighted). We compile basic statistics of these networks in the Technical Appendix, and present a quantitative overview of our results in Fig. 3. We observe that, in line with expectations derived from theory, more graphs or graphs with more potential edges, partitioned into fewer groups, generally yield more patterns.
Functional Brain Networks
Network neuroscience has emerged as a promising approach to understanding neurological disorders and diseases Bullmore and Sporns (2009); Bassett and Sporns (2017); Fornito, Zalesky, and Breakspear (2015). One of its fundamental questions is whether certain disorders are systematically associated with structural or functional connectivity alterations in the brain van den Heuvel and Sporns (2019). In particular, there is considerable uncertainty surrounding the neurological footprint of autism (and the delineation of its subtypes), and small sample sizes as well as covariates make many published findings hard to replicate He, Byrge, and Kennedy (2020); King et al. (2019). This calls for methods that can detect signal in the presence of considerable noise and heterogeneity, identifying connectivity patterns that are statistically significantly associated with one or more groups of brain networks.
Motivated by this application, we obtain graphs from preprocessed functional connectomes provided by the Autism Brain Imaging Data Exchange Craddock et al. (2013). In these graphs, each node corresponds to one of the regions of interest from the automated anatomical labeling atlas (AAL, Rolls et al., 2020), and each edge indicates relatively strong connectivity between two regions, as measured by their bloodoxygenlevel dependent signal correlation during restingstate functional magnetic resonance imaging. To facilitate comparisons, the data is processed and grouped as described by Lanciano, Bonchi, and Gionis (2020), but we remove the selfloops (corresponding to perfect selfcorrelations) that are present in their data.
We experiment with Gragra in four twogroup settings (individuals with autism spectrum disorder [ASD] and typically developed controls [TD] in the categories adolescents, children, eyes closed during scan, and males), four onegroup settings (autistic individuals in each category only), and one fourgroup setting (autistic and nonautistic children and adolescents), operating on graphs with edges and graph groups with graphs. Our fourgroup experiment identifies significant overconnectivity across multiple brain regions as characteristic of ASD children versus all other groups, paralleling the neuroscience literature Supekar et al. (2013); Nomi and Uddin (2015). However, as shown in Fig. 4, most of the patterns we identify in the twogroup setting yield similar information gains across both groups (left), and there is significant structure to be exploited even within individual groups (right). This indicates that the differences between autistic and nonautistic brains in the settings under study are rather subtle, and that there is considerable heterogeneity also in the onegroup data. To explore this heterogeneity and delineate neurosubtypes of autism (cf. Hong et al., 2020), our results could be used as inputs to multivariate subgroup discovery or clustering algorithms, where Gragra would effectively serve as a dimensionality reduction technique.
Air Transportation Networks
We obtain data on passenger flows between domestic airports in the United States for each month over the sixteen years from January to December from the website of the Bureau of Transportation Statistics Bureau of Transportation Statistics (2021)
. Restricting our analysis to United States mainland airports and carriers classified as national (
million to billion USD revenue in the previous year) or major (over billion USD revenue in the previous year), we create one air transportation network per year, month, and carrier class. To this end, for each year and month, we aggregate the passenger flows between two airports by carrier class and filter edges corresponding to fewer than passengers, which leaves edges between airports (identified by threeletter IATA codes). Excluding graphs with fewer than edges, we arrive at graphs, whose edges we discretize into ten weight categories using equalwidth binning.We are interested in discovering patterns that are shared across all graphs, identifying structures of connected routes that are specific to individual carrier classes, and unveiling both seasonal and temporal trends. Therefore, we run Gragra in six different settings: on all graphs as one group, on the graphs corresponding to each carrier class separately, on all graphs with carrier classes as groups, on all graphs with quarters as groups (starting from December to capture the winter holiday season), and on all graphs with consecutive fouryear intervals as groups. Thus, our setup contains graphs with edges and graph groups with graphs. In Fig. 5, we depict a subset of our results from the experiments involving the distinction between carrier classes. Gragra reveals an air transportation backbone jointly serviced by both carrier classes (middle), and it uncovers routes that are characteristically served by national or major carriers (left and right). Overall, we find that patterns corresponding to national carrier routes are often smaller and cover shorter distances than those corresponding to major carrier routes, mirroring the relatively smaller role of national carriers in the air traffic market.
International Trade Networks
We obtain data on international trade flows from the website of the World Integrated Trade Solution World Bank (2021), for the thirty years from to (inclusive). The raw data correspond to exports of goods between (mostly) countries, classified using the Harmonized System at the fourdigit level (HS4), whose trade values we aggregate per (source, destination, HS4 code) triple. For each year and HS4 code, we construct one directed, weighted graph with (roughly) countries as nodes and exports as edges, discretizing the edge weights into ten categories using equalwidth binning. We eliminate all trade entities above the country level but retain trade entities below the country level (and countries that do not exist anymore) if they have an ISO3 code. Restricting our attention to the WITS product groups Animals, Vegetables, Food Products, Minerals, and Chemicals, we arrive at graphs with nodes and at least edges.
Leveraging the richness of our data, we ask not only what graph patterns are characteristic of international trade as a whole, but also what structures emerge when we group trade networks by product class, tenyear interval, or product class and tenyear interval. As Gragra allows us to inspect our data at different scales, we further investigate the trade patterns it unveils when considering each product class separately, either treating all graphs from one product class as one group or splitting them by tenyear interval. Thus, we run our experiments on graphs with edges and graph groups with graphs. In Fig. 6, we illustrate five patterns discovered in the experiments that explore all graphs together, grouped by product class and tenyear interval. Although the input consists of fifteen classes, Gragra discovers not only meaningful patterns but meaningful patterns with meaningful assignments to graph groups which, as highlighted by the pattern labels in Fig. 6, can be summarized succinctly. Across all experiments, we observe that the patterns yielding the largest information gains are often composed entirely of edges in the top two weight bins. This suggests that the ranking of exporterimporter pairs is most stable on the upper end of the tradevalue spectrum, which aligns with interdisciplinary research findings that international trade is highly stratified Sacks, Ventresca, and Uzzi (2001); Lloyd, Mahutga, and De Leeuw (2009); Fagiolo, Reyes, and Schiavo (2010).
Conclusion
We study the graph group analysis problem: Given a set of graphs and a partition of this set into graph groups, succinctly summarize the commonalities and differences between graphs in the same group, between graphs in different groups, and between the relationships connecting the groups. We introduce Gragra as an algorithm to solve the problem, which uses maximum likelihood modeling, paired with a model selection criterion and a statistical test, to jointly discover a set of significant subgraphs, called graph patterns, and an assignment of these patterns to graph groups. In our experiments, we demonstrate that Gragra differentially describes synthetic and realworld graph groups, even when faced with heterogeneity, noise, or large group numbers. As a byproduct, we introduce two novel datasets of nodealigned graphs, which might be of independent interest to the graph mining community.
However, our work also has limitations. First of all, we model edge weights as categories, which works well for binned edge weights in practice but is theoretically dissatisfying. Therefore, a natural enhancement of Gragra would be able to handle real edge weights, possibly using a maximum entropy model on its edge weight distribution. Second, we currently test all our graph patterns at the same alpha level. While this is theoretically defensible, given that we combine our statistical test with a model selection criterion, dynamically adjusting our alpha level might be an option worth exploring. Finally, Gragra is currently limited to groups of nodealigned graphs, and extending it to other graph types constitutes an open opportunity for future work.
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